#8712701

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#58205	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,10,8,4)(5,9,15,16)(11,14,13,12)(17,18)(19,24)(20,22)(21,30)(23,25)(26,28)(27,31)(29,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(2,7)(3,8)(9,16)(11,13)(17,19,23,27)(18,24,25,31)(20,21,29,28)(22,30,32,26)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(2,7)(3,8)(9,16)(11,13)(18,25)(19,27)(20,29)(26,30)(35,44)(36,45)(40,46)(41,48);(1,3,6,8)(2,4,7,10)(5,13,15,11)(9,14,16,12)(17,20,23,29)(18,22,25,32)(19,28,27,21)(24,26,31,30)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21,23,28)(18,26,25,30)(19,29,27,20)(22,31,32,24)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,5,6,15)(2,9,7,16)(3,11,8,13)(4,12,10,14)(17,22)(18,20)(19,30)(21,24)(23,32)(25,29)(26,27)(28,31)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8619897);